Online Model Selection Based on the Variational Bayes

<strong>Online</strong> <strong>Model</strong> <strong>Selection</strong> <strong>Based</strong> on the Variational Bayes 1663
D 1
c g(t ) ± h
TEz r(x(t ), z(t ))| µ N (t ) i
C c 0®0 ¡ c ® (t ¡ 1) ² . (3.11)
By using equation 3.7, the ensemble average of the parameter N µ (t ) de�ned
in equation 3.6 can be calculated as
µ
N (t ) D hµi ® (t ¡1) D 1 @©
c @® (® (t ¡ 1), c ). (3.12)
The online VB algorithm is summarized as follows. In the VB E-step, the
ensemble average of the parameter N µ (t ) is determined by equation 3.12. Using
this value, one calculates the expectation value of the suf�cient statistics
(see equation 3.10) for the current data. The posterior hyperparameter ® (t )
is updated by equation 3.11 in the VB M-step. This process is repeated when
new data are observed. By combining the VB E-step (3.12) and VB M-step
(3.11) equations, one can get the recursive update equation for ® (t ):
D® (t ) D 1
c g(t
�
£
) TEz r(x(t ), z(t ))|hh i ® (t ¡1) ¤
C c 0®0 ¡ c ® (t ¡ 1)
´
. (3.13)
3.3 Stochastic Approximation. Unlike the VB algorithm, the discounted
free energy in the online VB algorithm does not always increase, because
a new contribution is added to the discounted free energy at each time
instance. In the following, we prove that the online VB algorithm can be
considered as a stochastic approximation (Kushner & Yin, 1997) for �nding
the maximum of the expected free energy de�ned in equation 3.2, which
gives a lower bound for the expected log evidence de�ned in equation 3.1.
The expected free energy (see equation 3.2), in which the maximization with
respect to Qz has been performed, can be written as
where
max E [F(XfTg, Qh , Qz)] r D E [FM(x, ®, T)] r , (3.14)
Qz
Z
FM(x, ®, T) D T
Z
£
dm (z)P(z|x, hµi ® )
dm (µ )Q h (µ ) log ¡ P(x, z| µ )/P(z|x, hµi ® ) ¢